Animation extends beyond last keyframe
It looks like a few objects in the model had "mesh cache modifiers" which were affecting the geometry in order to create the animation.

I followed the instructions on this answer to "bake" the mesh cache into regular animation steps. I used this .pc2 importer add-on to use the point caches that had been provided to me; exporting them to .mdd and then importing again did not work.

Once that was done, I exported to a COLLADA file but it was huge—80 MB. I applied this trick to merge animations into one. But the animation does not work yet.